Model Reduction in Control and Simulation: Algorithms and

Model Reduction in Control and Simulation:Algorithms and Applications

Peter Benner

Professur Mathematik in Industrie und TechnikFakultat fur Mathematik

Technische Universitat Chemnitz

Oxford UniversityComputational Mathematics and Applications Seminar

18 October 2007

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Current andFuture Work

References

Overview

1 IntroductionModel ReductionModel Reduction for Linear SystemsApplication Areas

2 Model ReductionGoalsModal TruncationPade ApproximationBalanced Truncation

3 ExamplesOptimal Control: Cooling of Steel ProfilesMicrothrusterButterfly GyroInterconnectReconstruction of the State

4 Current and Future Work

5 References

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples


References

Joint work with

Enrique Quintana-Ortı, Gregorio Quintana-Ortı, RafaMayo, Jose Manuel Badıa, Alfredo Remon, SergioBarrachina (Universidad Jaume I de Castellon, Spain).

Ulrike Baur, Matthias Pester, Jens Saak( )

.

Viatcheslav Sokolov(

former).

Heike Faßbender (TU Braunschweig).

Infineon Technologies/Qimonda, IMTEK (U Freiburg),iwb (TU Munchen), . . .

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Introduction

Problem

Given a physical problem with dynamics described by the statesx ∈ Rn, where n is the dimension of the state space.

Because of redundancies, complexity, etc., we want to describe thedynamics of the system using a reduced number of states.

This is the task of model reduction (also: dimension reduction, orderreduction).

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Introduction

Problem




Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Introduction

Problem




Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Motivation: Image Compression by Truncated SVD

A digital image with nx × ny pixels can be represented as matrixX ∈ Rnx×ny , where xij contains color information of pixel (i , j).

Memory: 4 · nx · ny bytes.

Theorem: (Schmidt-Mirsky/Eckart-Young)

Best rank-r approximation to X ∈ Rnx×ny w.r.t. spectral norm:

X =∑r

j=1σjujv

Tj ,

where X = UΣV T is the singular value decomposition (SVD) of X .

The approximation error is ‖X − X‖2 = σr+1.

Idea for dimension reduction

Instead of X save u1, . . . , ur , σ1v1, . . . , σrvr . memory = 4r × (nx + ny ) bytes.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References






X =∑r

j=1σjujv

Tj ,





Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References






X =∑r

j=1σjujv

Tj ,





Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Example: Image Compression by Truncated SVD

Example: Clown

320× 200 pixel ≈ 256 kb

rank r = 50, ≈ 104 kb

rank r = 20, ≈ 42 kb

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References


Example: Clown

320× 200 pixel ≈ 256 kb

rank r = 50, ≈ 104 kb

rank r = 20, ≈ 42 kb

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References


Example: Clown

320× 200 pixel ≈ 256 kb

rank r = 50, ≈ 104 kb

rank r = 20, ≈ 42 kb

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Dimension Reduction via SVD

Example: Gatlinburg

Organizing committeeGatlinburg/Householder Meeting 1964:

James H. Wilkinson, Wallace Givens,

George Forsythe, Alston Householder,

Peter Henrici, Fritz L. Bauer.

640× 480 pixel, ≈ 1229 kb

rank r = 100, ≈ 448 kb

rank r = 50, ≈ 224 kb

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References


Example: Gatlinburg





640× 480 pixel, ≈ 1229 kb

rank r = 100, ≈ 448 kb

rank r = 50, ≈ 224 kb

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References


Example: Gatlinburg





640× 480 pixel, ≈ 1229 kb

rank r = 100, ≈ 448 kb

rank r = 50, ≈ 224 kb

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Background

Image data compression via SVD works, if the singular values decay(exponentially).

Singular Values of the Image Data Matrices

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

The Model Reduction Problem

Dynamical Systems

Σ :

x(t) = f (t, x(t), u(t)), x(t0) = x0,y(t) = g(t, x(t), u(t))

with

states x(t) ∈ Rn,

inputs u(t) ∈ Rm,

outputs y(t) ∈ Rp.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Model Reduction for Dynamical Systems

Original System

Σ :

x(t) = f (t, x(t), u(t)),y(t) = g(t, x(t), u(t)).

states x(t) ∈ Rn,

inputs u(t) ∈ Rm,


Reduced-Order System

bΣ :

˙x(t) = bf (t, x(t), u(t)),y(t) = bg(t, x(t), u(t)).

states x(t) ∈ Rr , r n

inputs u(t) ∈ Rm,


Goal:

‖y − y‖ < tolerance · ‖u‖ for all admissible input signals.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References


Original System

Σ :


states x(t) ∈ Rn,

inputs u(t) ∈ Rm,



bΣ :



inputs u(t) ∈ Rm,


Goal:


Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References


Original System

Σ :


states x(t) ∈ Rn,

inputs u(t) ∈ Rm,



bΣ :



inputs u(t) ∈ Rm,


Goal:


Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Model Reduction for Linear Systems

Linear, Time-Invariant (LTI) Systems

x(t) = f (t, x , u) = Ax + Bu, A ∈ Rn×n, B ∈ Rn×m,y(t) = g(t, x , u) = Cx + Du, C ∈ Rp×n, D ∈ Rp×m.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References




State-Space Description for I/O-Relation (D = 0)

S : u 7→ y , y(t) = (h ? u)(t) =

∫ ∞−∞

h(t − τ)u(τ) dτ,

where h(s) =

CeAsB if s > 00 if s ≤ 0

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References





S : u 7→ y , y(t) = (h ? u)(t) =

∫ ∞−∞

h(t − τ)u(τ) dτ,

where h(s) =

CeAsB if s > 00 if s ≤ 0

Note: operator S not suitable for approximation as singular values arecontinuous; for model reduction use Hankel operator H.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References





H : u− 7→ y+, y+(t) =∫ 0

−∞ CeA(t−τ)Bu(τ) dτ for all t > 0.


Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References





H : u− 7→ y+, y+(t) =∫ 0



H compact ⇒ H has discrete SVD Hankel singular values

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References





H : u− 7→ y+, y+(t) =∫ 0


H compact ⇒H has discrete SVD Hankel singular values

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References





H : u− 7→ y+, y+(t) =∫ 0


H compact ⇒ H has discrete SVD

⇒ Best approx. problem w.r.t. 2-induced operator norm(Hankel norm) well-posed.

⇒ solution: Adamjan-Arov-Krein (AAK Theory, 1971/78).

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References





H : u− 7→ y+, y+(t) =∫ 0


H compact ⇒ H has discrete SVD

⇒ Best approx. problem w.r.t. 2-induced operator norm(Hankel norm) well-posed.

⇒ solution: Adamjan-Arov-Krein (AAK Theory, 1971/78).

But: computationally unfeasible for large-scale systems.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Linear Systems in Frequency Domain


f (t, x , u) = Ax + Bu, A ∈ Rn×n, B ∈ Rn×m,g(t, x , u) = Cx + Du, C ∈ Rp×n, D ∈ Rp×m.

Laplace Transformation / Frequency Domain

Application of Laplace transformation (x(t) 7→ x(s), x(t) 7→ sx(s))to linear system with x(0) = 0:

sx(s) = Ax(s) + Bu(s), y(s) = Bx(s) + Du(s),

yields I/O-relation in frequency domain:

y(s) =(

C (sIn − A)−1B + D︸︷︷︸=:G(s)

)u(s)

G is the transfer function of Σ.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References








y(s) =(

C (sIn − A)−1B + D︸︷︷︸=:G(s)

)u(s)


Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References








y(s) =(

C (sIn − A)−1B + D︸︷︷︸=:G(s)

)u(s)


Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References


Problem

Approximate the dynamical system

x = Ax + Bu, A ∈ Rn×n, B ∈ Rn×m,y = Cx + Du, C ∈ Rp×n, D ∈ Rp×m,

by reduced-order system

˙x = Ax + Bu, A ∈ Rr×r , B ∈ Rr×m,

y = C x + Du, C ∈ Rp×r , D ∈ Rp×m,

of order r n, such that

‖y − y‖ = ‖Gu − Gu‖ ≤ ‖G − G‖‖u‖ < tolerance · ‖u‖.

=⇒ Approximation problem: minorder (G)≤r ‖G − G‖.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References


Problem

Approximate the dynamical system

x = Ax + Bu, A ∈ Rn×n, B ∈ Rn×m,y = Cx + Du, C ∈ Rp×n, D ∈ Rp×m,

by reduced-order system

˙x = Ax + Bu, A ∈ Rr×r , B ∈ Rr×m,

y = C x + Du, C ∈ Rp×r , D ∈ Rp×m,

of order r n, such that

‖y − y‖ = ‖Gu − Gu‖ ≤ ‖G − G‖‖u‖ < tolerance · ‖u‖.

=⇒ Approximation problem: minorder (G)≤r ‖G − G‖.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Application Areas(Optimal) Control

Feedback Controllers

A feedback controller (dynamiccompensator) is a linear system oforder N, where

input = output of plant,

output = input of plant.

Modern (LQG-/H2-/H∞-) controldesign: N ≥ n

⇒ reduce order of original system.

Real-time control is only possible with controllers of low complexity.

Experience tells us: the more complex, the more fragile.

Modern feedback control of systems governed by PDEs impossibledue to large scale of systems arising from FE discretization.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References











Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References











Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References











Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References











Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Application AreasMicro Electronics

Progressive miniaturization: Moore’s Law states that thenumber of on-chip transistors doubles each 12 (now: 18) months.

Verification of VLSI/ULSI chip design requires high number ofsimulations for different input signals.

Increase in packing density requires modeling of interconncet toensure that thermic/electro-magnetic effects do not disturbsignal transmission.

Linear systems in micro electronics occur through modifiednodal analysis (MNA) for RLC networks, e.g., when

– decoupling large linear subcircuits,– modeling transmission lines,– modeling pin packages in VLSI chips,– modeling circuit elements described by Maxwell’s equation

using partial element equivalent circuits (PEEC).

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References








Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References








Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References








Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Application AreasMicro Electronics: Example for Miniaturization

Intel 4004 (1971)

1 layer, 10µ technology,

2,300 transistors,

64 kHz clock speed.

Intel Pentium IV (2001)

7 layers, 0.18µ technology,

42,000,000 transistors,

2 GHz clock speed,

2km of interconnect.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References

Application AreasMEMS/Microsystems

Typical problem in MEMS simulation:coupling of different models (thermic, structural, electric,electro-magnetic) during simulation.

Problems and Challenges:

Reduce simulation times by replacing sub-systems with theirreduced-order models.

Stability properties of coupled system may deteriorate throughmodel reduction even when stable sub-systems are replaced bystable reduced-order models.

Multi-scale phenomena.

Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References







Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References







Model Reduction

Peter Benner

Introduction

Model Reduction

Linear Systems

ApplicationAreas

Model Reduction

Examples


References







Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Model ReductionGoals

Automatic generation of compact models.

Satisfy desired error tolerance for all admissible input signals,i.e., want

‖y − y‖ < tolerance · ‖u‖ ∀u ∈ L2(R,Rm).

=⇒ Need computable error bound/estimate!

Preserve physical properties:

– stability (poles of G in C−),– minimum phase (zeroes of G in C−),– passivity (“system does not generate energy”).

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References








Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References








Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References








Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References








Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References








Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Model Reduction

1 Modal Truncation

2 Pade-Approximation and Krylov Subspace Methods

3 Balanced Truncation

4 many more. . .

Joint feature of many methods: Galerkin or Petrov-Galerkin-typeprojection of state-space onto low-dimensional subspace V along W:assume x ≈ VW T x =: x , where

range (V ) = V, range (W ) =W, W TV = Ir .

Then, with x = W T x , we obtain x ≈ V x and

‖x − x‖ = ‖x − V x‖.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Model Reduction

1 Modal Truncation

2 Pade-Approximation and Krylov Subspace Methods

3 Balanced Truncation

4 many more. . .

Joint feature of many methods: Galerkin or Petrov-Galerkin-typeprojection of state-space onto low-dimensional subspace V along W:assume x ≈ VW T x =: x , where

range (V ) = V, range (W ) =W, W TV = Ir .

Then, with x = W T x , we obtain x ≈ V x and

‖x − x‖ = ‖x − V x‖.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Modal Truncation

Idea:

Project state-space onto A-invariant subspace V, where

V = span(v1, . . . , vr ),

vk = eigenvectors corresp. to “dominant” modes ≡ eigenvalues of A.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Modal Truncation

Idea:


V = span(v1, . . . , vr ),


Properties:

Simple computation for large-scale systems, using, e.g., Krylovsubspace methods (Lanczos, Arnoldi), Jacobi-Davidson method.

Error bound:

‖G − G‖∞ ≤ cond2 (T ) ‖C2‖2‖B2‖21

minλ∈Λ (A2) |Re(λ)|,

where T−1AT = diag(A1,A2).

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Modal Truncation

Idea:


V = span(v1, . . . , vr ),


Difficulties:

Eigenvalues contain only limited system information.

Dominance measures are difficult to compute.(Litz 1979: use Jordan canoncial form; otherwise merelyheuristic criteria.Rommes 2007: dominant pole algorithm (two-sided RQI).)

Error bound not computable for really large-scale probems.

New direction: AMLS (automated multilevel substructuring)[Benninghof/Lehoucq ’04, Elssel/Voß ’05, Blomeling ’06].

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation

Idea:

ConsiderEx = Ax + Bu, y = Cx

with rational transfer function G (s) = C (sE − A)−1B.

For s0 6∈ Λ (A,E ):

G (s) = m0 + m1(s − s0) + m2(s − s0)2 + . . .

As reduced-order model use rth Pade approximant G to G :

G (s) = G (s) +O((s − s0)2r ),

i.e., mj = mj for j = 0, . . . , 2r − 1

moment matching if s0 <∞,

partial realization if s0 =∞.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation

Idea:



For s0 6∈ Λ (A,E ):

G (s) = m0 + m1(s − s0) + m2(s − s0)2 + . . .


G (s) = G (s) +O((s − s0)2r ),

i.e., mj = mj for j = 0, . . . , 2r − 1



Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation

Idea:



For s0 6∈ Λ (A,E ):

G (s) = m0 + m1(s − s0) + m2(s − s0)2 + . . .


G (s) = G (s) +O((s − s0)2r ),

i.e., mj = mj for j = 0, . . . , 2r − 1



Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation

Pade-via-Lanczos Method (PVL)

Moments need not be computed explicitly; moment matching isequivalent to projecting state-space onto

V = span(B, AB, . . . , Ar−1B) = K(A, B, r)

(where A = (s0E − A)−1E , B = (s0E − A)−1B) along

W = span(CH , AHCH , . . . , (AH)r−1CH) = K(AH ,CH , r).

Computation via unsymmetric Lanczos method, yields systemmatrices of reduced-order model as by-product.

PVL applies w/o changes for singular E if s0 6∈ Λ (A,E ):– for s0 6=∞: Gallivan/Grimme/Van Dooren 1994,

Freund/Feldmann 1996, Grimme 1997

– for s0 =∞: B./Sokolov 2005

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation










Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation










Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation


Partial realization for descriptor systems: [B./Sokolov, SCL, 2006]

For nonsingular E and s0 =∞:

moments = Markov parameters = C (E−1A)jE−1B, j = 0, 1, . . .

Question: for E singular, what is the correct generalized inverse here?

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation






Answer: 2-inverse E2 = Q

[Inf

00 0

]P, where

sPEQ − PAQ = s

[Inf

00 N

]−[

J 00 In∞

],

is the Weierstraß canonical form (WCF) of sE − A.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation






Answer: 2-inverse E2 = Q

[Inf

00 0

]P, where

sPEQ − PAQ = s

[Inf

00 N

]−[

J 00 In∞

],

is the Weierstraß canonical form (WCF) of sE − A.

P,Q can be computed w/o WCF in many applications.

Numerically, use Lanczos applied to E2A,E2B,C.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation


Difficulties:

No computable error estimates/bounds for ‖y − y‖2.

Mostly heuristic criteria for choice of expansion points.Optimal choice for second-order systems with proportional/Rayleigh

damping (Beattie/Gugercin 2005).

Good approximation quality only locally.

Preservation of physical properties only in very special cases;usually requires post processing which (partially) destroysmoment matching properties.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation


Difficulties:






Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation


Difficulties:






Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation


Difficulties:






Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation


Difficulties:






New direction: moment matching yields rational interpolation ofG (j)(s) for j = 0, . . . , 2r − 1 at s = s0.Instead: use rational (Hermite) interpolation at sj , j = 0, . . . , r .

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Pade Approximation


Difficulties:






New direction: moment matching yields rational interpolation ofG (j)(s) for j = 0, . . . , 2r − 1 at s = s0.Instead: use rational (Hermite) interpolation at sj , j = 0, . . . , r .Current work: where to put the sj?

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Idea:

A system Σ, realized by (A,B,C ,D), is called balanced, ifsolutions P,Q of the Lyapunov equations

AP + PAT + BBT = 0, ATQ + QA + CTC = 0,

satisfy: P = Q = diag(σ1, . . . , σn) with σ1 ≥ σ2 ≥ . . . ≥ σn > 0.

σ1, . . . , σn are the Hankel singular values (HSVs) of Σ.

Compute balanced realization of the system via state-spacetransformation

T : (A,B,C ,D) 7→ (TAT−1,TB,CT−1,D)

=

„»A11 A12

A21 A22

–,

»B1

B2

–,ˆ

C1 C2

˜,D

«Truncation (A, B, C , D) = (A11,B1,C1,D).

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Idea:






T : (A,B,C ,D) 7→ (TAT−1,TB,CT−1,D)

=

„»A11 A12

A21 A22

–,

»B1

B2

–,ˆ

C1 C2

˜,D


Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Idea:






T : (A,B,C ,D) 7→ (TAT−1,TB,CT−1,D)

=

„»A11 A12

A21 A22

–,

»B1

B2

–,ˆ

C1 C2

˜,D


Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Idea:






T : (A,B,C ,D) 7→ (TAT−1,TB,CT−1,D)

=

„»A11 A12

A21 A22

–,

»B1

B2

–,ˆ

C1 C2

˜,D


Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Motivation:

HSV are system invariants: they are preserved under T and determinethe energy transfer given by the Hankel map

H : L2(−∞, 0) 7→ L2(0,∞) : u− 7→ y+.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Motivation:


H : L2(−∞, 0) 7→ L2(0,∞) : u− 7→ y+.

In balanced coordinates . . . energy transfer from u− to y+:

E := supu∈L2(−∞,0]

x(0)=x0

∞∫0

y(t)T y(t) dt

0∫−∞

u(t)Tu(t) dt

=1

‖x0‖2

n∑j=1

σ2j x

20,j

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Motivation:


H : L2(−∞, 0) 7→ L2(0,∞) : u− 7→ y+.

In balanced coordinates . . . energy transfer from u− to y+:

E := supu∈L2(−∞,0]

x(0)=x0

∞∫0

y(t)T y(t) dt

0∫−∞

u(t)Tu(t) dt

=1

‖x0‖2

n∑j=1

σ2j x

20,j

=⇒ Truncate states corresponding to “small” HSVs=⇒ complete analogy to best approximation via SVD!

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Implementation: SR Method

1 Compute (Cholesky) factors of the solutions of the Lyapunovequations,

P = STS , Q = RTR.

2 Compute SVD

SRT = [ U1, U2 ]

[Σ1

Σ2

] [V T

1

V T2

].

3 SetW = RTV1Σ

−1/21 , V = STU1Σ

−1/21 .

4 Reduced model is (W TAV ,W TB,CV ,D).

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation



P = STS , Q = RTR.

2 Compute SVD

SRT = [ U1, U2 ]

[Σ1

Σ2

] [V T

1

V T2

].

3 SetW = RTV1Σ

−1/21 , V = STU1Σ

−1/21 .


Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation



P = STS , Q = RTR.

2 Compute SVD

SRT = [ U1, U2 ]

[Σ1

Σ2

] [V T

1

V T2

].

3 SetW = RTV1Σ

−1/21 , V = STU1Σ

−1/21 .


Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Properties:

Reduced-order model is stable with HSVs σ1, . . . , σr .

Adaptive choice of r via computable error bound:

‖y − y‖2 ≤(

2∑n

k=r+1σk

)‖u‖2.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Properties:

Reduced-order model is stable with HSVs σ1, . . . , σr .

Adaptive choice of r via computable error bound:

‖y − y‖2 ≤(

2∑n

k=r+1σk

)‖u‖2.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Properties:

General misconception: complexity O(n3) – true for severalimplementations! (e.g., Matlab, SLICOT, MorLAB).

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Properties:


New algorithmic ideas from numerical linear algebra:

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Properties:



– Instead of Gramians P,Qcompute S ,R ∈ Rn×k ,k n, such that

P ≈ SST , Q ≈ RRT .

– Compute S ,R withproblem-specific Lyapunovsolvers of “low” complexitydirectly.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Properties:



Parallelization:

– Efficient parallel algorithms based on matrix sign function.

– Complexity O(n3/q) on q-processor machine.

– Software library PLiCMR with WebComputing interface.

(B./Quintana-Ortı/Quintana-Ortı since 1999)

Formatted Arithmetic:

For special problems from PDE control use implementation based onhierarchical matrices and matrix sign function method (Baur/B.),complexity O(n log2(n)r2).

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Properties:



Parallelization:

– Efficient parallel algorithms based on matrix sign function.

– Complexity O(n3/q) on q-processor machine.

– Software library PLiCMR with WebComputing interface.

(B./Quintana-Ortı/Quintana-Ortı since 1999)

Formatted Arithmetic:

For special problems from PDE control use implementation based onhierarchical matrices and matrix sign function method (Baur/B.),complexity O(n log2(n)r2).

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation

Properties:



Sparse Balanced Truncation:

– Sparse implementation using sparse Lyapunov solver(ADI+MUMPS/SuperLU).

– Complexity O(n(k2 + r2)).

– Software:

+ Matlab toolbox LyaPack (Penzl 1999),+ Software library SpaRed with WebComputing interface.

(Badıa/B./Quintana-Ortı/Quintana-Ortı since 2003)

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References

Balanced Truncation for Unstable Systems

For A stable, Gramians are defined by

P =

∫ ∞0

eAtBBT eAT t dt, Q =

∫ ∞0

eAT tCTCeAt dt.

For unstable A, integrals diverge!

Frequency-domain definition of Gramians

P := 12π

∫∞−∞(jω − A)−1BBT (jω − A)−H dω,

Q := 12π

∫∞−∞(jω − A)−HCTC (jω − A)−1 dω.

Well-defined if Λ (A) ∩ ıR = ∅!For stable A, definitions coincide.

Balancing/balanced truncation can be based on P, Q.Moreover, BT error bound holds!

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References



P =

∫ ∞0


∫ ∞0

eAT tCTCeAt dt.



P := 12π

∫∞−∞(jω − A)−1BBT (jω − A)−H dω,

Q := 12π

∫∞−∞(jω − A)−HCTC (jω − A)−1 dω.



Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References



P =

∫ ∞0


∫ ∞0

eAT tCTCeAt dt.



P := 12π

∫∞−∞(jω − A)−1BBT (jω − A)−H dω,

Q := 12π

∫∞−∞(jω − A)−HCTC (jω − A)−1 dω.



Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References



P =

∫ ∞0


∫ ∞0

eAT tCTCeAt dt.



P := 12π

∫∞−∞(jω − A)−1BBT (jω − A)−H dω,

Q := 12π

∫∞−∞(jω − A)−HCTC (jω − A)−1 dω.



Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References



P =

∫ ∞0


∫ ∞0

eAT tCTCeAt dt.



P := 12π

∫∞−∞(jω − A)−1BBT (jω − A)−H dω,

Q := 12π

∫∞−∞(jω − A)−HCTC (jω − A)−1 dω.



Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References


Computation of Unstable Gramians

If (A,B) stabilizable, (A,C ) detectable, and Λ (A) ∩ ıR = ∅, thenP,Q are solutions of the Lyapunov equations

(A− BBTX )P + P(A− BBTX )T + BBT = 0,

(A− YCTC )TQ + Q(A− YCTC ) + CTC = 0,

where X and Y are the stabilizing solutions of the dual algebraicBernoulli equations

ATX + XA− XBBTX = 0,

AY + YAT − YCTCY = 0.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References


Theorem [B. 2006]

Let (A,B) be stabilizable, Λ (A)∩ ıR = ∅, and X+ be the unique stabilizingsolution of the ABE

ATX + XA− XBBTX = 0.

Then

a) rank (X+) = k, where k is the number of eigenvalues of A in C+.

b) A full-rank factor Y+ ∈ Rn×k of X+ is given by

Y+ =√

2QY R−1,

where colspan(QY ) is basis of anti-stable A-invariant subspace, R is

defined via sign“h

AT

0BBT

−A

i”.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References


Theorem [B. 2006]



Then



Y+ =√

2QY R−1,



AT

0BBT

−A

i”.

Model Reduction

Peter Benner

Introduction

Model Reduction

Goals

ModalTruncation

PadeApproximation

BalancedTruncation

Examples


References


Theorem [B. 2006]



Then



Y+ =√

2QY R−1,



AT

0BBT

−A

i”.

Efficient solution of ABEs: sign-function based computation of Y+

[Barrachina/B./Quintana-Ortı].Current work: solvers for large-scale ABEs with (data-)sparse A.

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect

Reconstructionof the State


References

ExamplesOptimal Control: Cooling of Steel Profiles

Mathematical model: boundary controlfor linearized 2D heat equation.

c · ρ ∂∂t

x = λ∆x , ξ ∈ Ω

λ∂

∂nx = κ(uk − x), ξ ∈ Γk , 1 ≤ k ≤ 7,

∂

∂nx = 0, ξ ∈ Γ7.

=⇒ m = 7, p = 6.

FEM Discretization, different modelsfor initial mesh (n = 371),1, 2, 3, 4 steps of mesh refinement ⇒n = 1357, 5177, 20209, 79841.

2

34

9 10

1516

22

34

43

47

51

55

60 63

8392

Source: Physical model: courtesy of Mannesmann/Demag.

Math. model: Troltzsch/Unger 1999/2001, Penzl 1999, Saak 2003.

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References


n = 1357, Absolute Error

– BT model computed with signfunction method,

– MT w/o static condensation,same order as BT model.

n = 79841, Absolute error

– BT model computed usingSpaRed,

– computation time: 8 min.

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References


n = 1357, Absolute Error

– BT model computed with signfunction method,

– MT w/o static condensation,same order as BT model.

n = 79841, Absolute error

– BT model computed usingSpaRed,

– computation time: 8 min.

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References

ExamplesMEMS: Microthruster

Co-integration of solid fuel withsilicon micromachined system.

Goal: Ignition of solid fuel cells byelectric impulse.

Application: nano satellites.

Thermo-dynamical model, ignitionvia heating an electric resistance byapplying voltage source.

Design problem: reach ignitiontemperature of fuel cell w/o firingneighbouring cells.

Spatial FEM discretizatoin ofthermo-dynamical model linearsystem, m = 1, p = 7.

Source: The Oberwolfach Benchmark Collection http://www.imtek.de/simulation/benchmark

Courtesy of C. Rossi, LAAS-CNRS/EU project “Micropyros”.

http://www.imtek.de/simulation/benchmark

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References


axial-symmetric 2D model

FEM discretisation using linear (quadratic) elements n = 4, 257(11, 445) m = 1, p = 7.

Reduced model computed using SpaRed. modal truncation usingARPACK, and Z. Bai’s PVL implementation.

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References





Relative error n = 4, 257

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References





Relative error n = 4, 257 Relative error n = 11, 445

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References





Frequency Response BT/PVL

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References





Frequency Response BT/PVL Frequency Response BT/MT

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References

ExamplesMEMS: Microgyroscope (Butterfly Gyro)

By applying AC voltage toelectrodes, wings are forced tovibrate in anti-phase in waferplane.

Coriolis forces induce motion ofwings out of wafer planeyielding sensor data.

Vibrating micro-mechanicalgyroscope for inertial navigation.

Rotational position sensor.

Source: The Oberwolfach Benchmark Collection http://www.imtek.de/simulation/benchmark

Courtesy of D. Billger (Imego Institute, Goteborg), Saab Bofors Dynamics AB.

http://www.imtek.de/simulation/benchmark

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References

ExamplesMEMS: Butterfly Gyro

FEM discretization of structure dynamical model using quadratictetrahedral elements (ANSYS-SOLID187) n = 34, 722, m = 1, p = 12.

Reduced model computed using SpaRed, r = 30.

Frequency Repsonse Analysis Hankel Singular Values

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References




Frequency Repsonse Analysis

Hankel Singular Values

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References




Frequency Repsonse Analysis Hankel Singular Values

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References

ExamplesMicro Electronics: Interconnect

RLC circuit, characteristic curve has falling edge at ω = 100 Hz.

n = 1999, m = p = 2, reduced model using PLiCMR: r = 20.

Accuracy of Reduced-Order Model

100

105

1010

From: In(2)

100

105

1010

−250

−200

−150

−100

−50

0

To: O

ut(2

)

−250

−200

−150

−100

−50

0From: In(1)

To: O

ut(1

)

Bode Diagram

Frequency (rad/sec)

Mag

nitu

de (d

B)

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References

ExamplesMicro Electronics: Interconnect

RLC circuit, characteristic curve has falling edge at ω = 100 Hz.

n = 1999, m = p = 2, reduced model using PLiCMR: r = 20.

Accuracy of Reduced-Order Model

100

105

1010

From: In(2)

100

105

1010

−250

−200

−150

−100

−50

0

To: O

ut(2

)

−250

−200

−150

−100

−50

0From: In(1)

To: O

ut(1

)

Bode Diagram

Frequency (rad/sec)

Mag

nitu

de (d

B)

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References

ExamplesReconstruction of the State

BT is often criticized for its bias towards the input-output behavior ofthe system. But states can also be reconstructed using

x(t) ≈ Vxr (t).

Example: 2D heat equation with localized heat source, 64× 64 grid,r = 6 model by BT, simulation for u(t) = 10 cos(t).

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References



x(t) ≈ Vxr (t).


Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References



x(t) ≈ Vxr (t).


Model Reduction

Peter Benner

Introduction

Model Reduction

Examples

Optimal Cooling

Microthruster

Butterfly Gyro

Interconnect



References

ExamplesBT modes are intelligent ansatz functions for Galerkin projection

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples


References

Current and Future Work

Parametric Models

x = A(p)x + B(p)u, y = C (p)x + D(p)u,

where p ∈ Rs are free parameters which should be preserved in thereduced-order model.

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples


References


Parametric Models

x = A(p)x + B(p)u, y = C (p)x + D(p)u,


Frequently: B,C ,D parameter independent,

A(p) = A0 + p1A1 + . . .+ psAs .

⇒ (Modified) linear model reduction methods applicable.

Multipoint expansion combined with Pade-type approx. possible.

New idea: BT for reference parameters combined withinterpolation yields parametric reduced-order models.

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples


References


Parametric Models

x = A(p)x + B(p)u, y = C (p)x + D(p)u,


Frequently: B,C ,D parameter independent,

A(p) = A0 + p1A1 + . . .+ psAs .

⇒ (Modified) linear model reduction methods applicable.

Multipoint expansion combined with Pade-type approx. possible.

New idea: BT for reference parameters combined withinterpolation yields parametric reduced-order models.

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples


References


Nonlinear Systems

Linear projection

x ≈ V x , ˙x = W T f (V x , u)

is in general not model reduction!

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples


References


Nonlinear Systems

Linear projection

x ≈ V x , ˙x = W T f (V x , u)


Need specific methods

– POD + balanced truncation empirical Gramians(Lall/Marsden/Glavaski 1999/2002),

– Approximate inertial manifold method (∼ staticcondensation for nonlinear systems).

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples


References


Nonlinear Systems

Linear projection

x ≈ V x , ˙x = W T f (V x , u)


Exploit structure of nonlinearities, e.g., in optimal control oflinear PDEs with nonlinear BCs

– bilinear control systems x = Ax +∑

j Njxuj + Bu,

– formal linear systems (cf. Follinger 1982)

x = Ax + N g(Hx) + Bu = Ax +[

B N] [ u

g(z)

],

where z := Hx ∈ R`, ` n.

Model Reduction

Peter Benner

Introduction

Model Reduction

Examples


References

References

1 G. Obinata and B.D.O. Anderson.Model Reduction for Control System Design.Springer-Verlag, London, UK, 2001.

2 Z. Bai.Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems.Appl. Numer. Math, 43(1–2):9–44, 2002.

3 R. Freund.Model reduction methods based on Krylov subspaces.Acta Numerica, 12:267–319, 2003.

4 P. Benner, E.S. Quintana-Ortı, and G. Quintana-Ortı.State-space truncation methods for parallel model reduction of large-scale systems.Parallel Comput., 29:1701–1722, 2003.

5 P. Benner, V. Mehrmann, and D. Sorensen (editors).Dimension Reduction of Large-Scale Systems.Lecture Notes in Computational Science and Engineering, Vol. 45,Springer-Verlag, Berlin/Heidelberg, Germany, 2005.

6 A.C. Antoulas.Lectures on the Approximation of Large-Scale Dynamical Systems.SIAM Publications, Philadelphia, PA, 2005.

7 P. Benner, R. Freund, D. Sorensen, and A. Varga (editors).Special issue on Order Reduction of Large-Scale Systems.Linear Algebra Appl., Vol. 415, Issues 2-3, pp. 231–578, June 2006.

Thanks for your attention!

Documents

Model Reduction in Control and Simulation: Algorithms and